The CPU has an embedded temperature sensor (within the package) to measure the temperature of the heat dissipation plate directly. The motherboard will panic and cut power to the system immediately if the CPU momentarily reaches 100 C. It is entirely possible to have this happen without melting wires or even having a case that feels abnormally warm.
The E7500 CPU came out in early 2009. If your system has been running fine for years (without changes) and spontaneously started suffering from heat issues: it's possible that your thermal paste has been drying out / breaking down slowly, and has finally degraded enough that sufficient heat is building up in your CPU to cause the motherboard to panic. If this is the case, you could just clean off (using rubbing alcohol and a q-tip) the old stuff and re-apply fresh thermal paste to your existing hardware.
If it turns out you need to buy something, a passive heatsink will not adequately dissipate the 65W coming from your CPU. An inexpensive heatsink & fan will though...
Here's a budget one ($10 delivered) with plenty of positive reviews:
Here's the one I would get ($15 delivered) if I wanted to keep the computer around for a while: